Finite Group Databasebeta

← C2×#23777406 #35196527

Subgroups — C2×#23777406 #35196527

Order 20224.

7854 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23777406C2132yesmaximal, direct product factor
2C2×#19099292C211yesmaximal, characteristic, complemented
3C2×#19099257C216yesmaximal, complemented
4C2×#19099235C211yesmaximal, characteristic, complemented
5C2×#19099234C213yesmaximal, complemented
6C2×#19099232C212yesmaximal, complemented
7C2×#19099218C213yesmaximal, complemented
8C2×#19099164C216yesmaximal, complemented
9C2×#19099113C213yesmaximal, complemented
10C2×#19099097C213yesmaximal, complemented
11C2×#19099045C212yesmaximal, complemented
12C2×C4×#41×C79C211yesmaximal, characteristic, complemented
13C22×#11929625C2212yes—
14C22×#11929624C2216yes—
15#19099292C22116yescomplemented
16#19099257C22196yescomplemented
17#19099235C22116yescomplemented
18#19099234C22148yescomplemented
19#19099232C22132yescomplemented
20#19099218C22148yescomplemented
21#19099164C22196yescomplemented
22#19099113C22148yescomplemented
23C2×#14579161C2213yes—
24C2×#14579158C2213yes—
25C2×#14579157C2216yes—
26#19099097C22148yescomplemented
27C2×#14579155C2216yes—
28C2×#14579153C2211yescharacteristic
29C2×#14579152C2216yes—
30C2×#14579149C2213yes—
31C2×#14579144C2216yes—
32C2×#14579143C22112yes—
33C2×#14579140C2213yes—
34C2×#14579136C2216yes—
35C2×#14579135C2216yes—
36C2×#14579135C2213yes—
37C2×#14579135C2216yes—
38C2×#14579134C22112yes—
39C2×#14579132C2216yes—
40#19099045C22132yescomplemented
41C2×Q8×Dic79C2212yes—
42C2×C4×Dic158C2213yes—
43C2×C4×D316C2216yes—
44C2×C4×D316C2213yes—
45C2×C4×D316C2216yes—
46C2×#14579123C2216yes—
47C2×C4×C22⋊D79C2216yes—
48C2×#14579118C2216yes—
49C2×C4⋊C4×D79C2216yes—
50C22×#41×C79C2211yescharacteristic
51C4×#41×C79C22116yescomplemented
52C2×C4×Q8×C79C2211yescharacteristic
53C2×C4×Q8×C79C2211yescharacteristic
54C2×C4×D4×C79C2213yes—
55C2×C4×D4×C79C2213yes—
56C2×#117×C79C2213yes—
57C22×C42×C79C2213yes—
58C2×#11929625—216no—
59C2×#11929625C23116yes—
60C2×#11929624—424no—
61C2×#11929624C23148yes—
62C22×Dic158C2313yes—
63C22×D316C2316yes—
64C22×D316C2313yes—
65C22×D316C2316yes—
66C22×C22⋊D79C2316yes—
67C22×C22⋊D79C23112yes—
68C22×C4×D79C2316yes—
69C22×C4×D79C23112yes—
70#14579161C23124yes—
71#14579158C23124yes—
72#14579157C23148yes—
73#14579155C23148yes—
74#14579153C2318yes—
75#14579152C23148yes—
76#14579149C23124yes—
77#14579144C23148yes—
78#14579143C23196yes—
79#14579140C23124yes—
80#14579136C23148yes—
81#14579135C23148yes—
82#14579135C23148yes—
83#14579135C23124yes—
84#14579134C23196yes—
85#14579132C23148yes—
86Q8×Dic79C23116yes—
87C4×Dic158C23124yes—
88C4×D316C23148yes—
89C4×D316C23124yes—
90C4×D316C23148yes—
91#14579123C23148yes—
92C4×C22⋊D79C23148yes—
93#14579118C23148yes—
94C2×C4⋊Dic79C2316yes—
95C2×C4⋊Dic79C2311yescharacteristic
96C2×C4⋊Dic79C2316yes—
97C2×C4⋊Dic79C2311yescharacteristic
98C2×Dic79⋊C4C2316yes—
99C2×Dic79⋊C4C2312yes—
100C2×Dic79⋊C4C2312yes—
101C2×Dic79⋊C4C2316yes—
102C4⋊C4×D79C23148yes—
103C2×C22⋊Dic79C2316yes—
104C2×(C2×D79)⋊C4C2316yes—
105C2×(C2×D79)⋊C4C2316yes—
106C2×(C2×D79)⋊C4C2316yes—
107C2×(C2×D79)⋊C4C2316yes—
108C2×C4×Dic79C2316yes—
109C2×#41×C79—28no—
110C2×#41×C79C2318yes—
111C22×Q8×C79C2311yescharacteristic
112C22×D4×C79C2313yes—
113C23×C4×C79C2313yes—
114C23×C4×C79C2313yes—
115C23×C4×C79C2313yes—
116C4×Q8×C79C2318yes—
117C4×Q8×C79C2318yes—
118C4×D4×C79C23124yes—
119C4×D4×C79C23124yes—
120#117×C79C23124yes—
121C2×C4⋊C4×C79C2313yes—
122C2×C4⋊C4×C79C2313yes—
123C2×C22⋊C4×C79C2316yes—
124C2×C42×C79C2311yescharacteristic
125C2×C42×C79C2313yes—
126C2×C42×C79C2313yes—
127C2×C42×C79C2313yes—
128C2×C42×C79C23124yes—
129C23×D79C2416yes—
130#11929625—264no—
131#11929624—496no—
132C2×Dic158—212no—
133C2×Dic158C24112yes—
134C2×D316—224no—
135C2×D316—212no—
136C2×D316—412no—
137C2×D316C24112yes—
138C2×D316C24124yes—
139C2×D316C24124yes—
140C2×C22⋊D79—424no—
141C2×C22⋊D79—412no—
142C2×C22⋊D79C24148yes—
143C2×C22⋊D79C24124yes—
144C22×Dic79C2416yes—
145C22×Dic79C2412yes—
146C22×Dic79C2412yes—
147C2×C4×D79—224no—
148C2×C4×D79—424no—
149C2×C4×D79C24124yes—
150C2×C4×D79C24148yes—
151C4⋊Dic79C24124yes—
152C4⋊Dic79C2414yes—
153C4⋊Dic79C24124yes—
154C4⋊Dic79C2414yes—
155Dic79⋊C4C24124yes—
156Dic79⋊C4C2418yes—
157Dic79⋊C4C2418yes—
158Dic79⋊C4C24124yes—
159C22⋊Dic79C24124yes—
160(C2×D79)⋊C4C24124yes—
161(C2×D79)⋊C4C24124yes—
162(C2×D79)⋊C4C24124yes—
163(C2×D79)⋊C4C24124yes—
164C4×Dic79C24124yes—
165C24×C79C2413yes—
166#41×C79—232no—
167C2×Q8×C79#4118yescomplemented
168C2×Q8×C79C2414yes—
169C2×D4×C79—212no—
170C2×D4×C79C24112yes—
171C22×C4×C79—212no—
172C22×C4×C79—212no—
173C22×C4×C79—46no—
174C22×C4×C79C24112yes—
175C22×C4×C79C2411yescharacteristic
176C22×C4×C79C2411yescharacteristic
177C22×C4×C79C2413yes—
178C22×C4×C79C2413yes—
179C22×C4×C79C24112yes—
180C22×C4×C79C24112yes—
181C22×C4×C79C2413yes—
182C22×C4×C79C2411yescharacteristic
183C4⋊C4×C79C24112yes—
184C4⋊C4×C79C24112yes—
185C22⋊C4×C79C24124yes—
186C42×C79C2414yes—
187C42×C79C24112yes—
188C42×C79C24112yes—
189C42×C79C24112yes—
190C23×C79—26no—
191C23×C79—26no—
192C23×C79—26no—
193C23×C79C2516yes—
194C23×C79C2511yescharacteristic
195C22×D79—212no—
196C22×D79—212no—
197C22×D79—212no—
198C22×D79C25112yes—
199Q8×C79C2×#41116yescomplemented
200D4×C79—224no—
201C2×C4×C79—22no—
202C2×C4×C79—26no—
203C2×C4×C79—26no—
204C2×C4×C79—22no—
205C2×C4×C79—224no—
206C2×C4×C79—224no—
207C2×C4×C79—412no—
208C2×C4×C79C2×#41112yes—
209C2×C4×C79H(5,𝔽2)14yes—
210C2×C4×C79C2512yes—
211C2×C4×C79C2512yes—
212C2×C4×C79C2516yes—
213C2×C4×C79C2516yes—
214C2×C4×C79C2516yes—
215C2×C4×C79C2512yes—
216C22⋊D79—424no—
217C22⋊D79—448no—
218C2×Dic79—24no—
219C2×Dic79—212no—
220C2×Dic79—24no—
221C2×Dic79C2514yes—
222C2×Dic79C25112yes—
223C2×Dic79C2514yes—
224D316—248no—
225D316—224no—
226D316—424no—
227C4×D79—248no—
228C4×D79—448no—
229Dic158—224no—
230C22×C79—26no—
231C22×C79—26no—
232C22×C79—26no—
233C22×C79—412no—
234C22×C79C22×#4112yes—
235C22×C79C2×H(5,𝔽2)12yes—
236C22×C79#58512yes—
237C22×C79C2611yescharacteristic
238C2×D79—212no—
239C2×D79—212no—
240C2×D79—212no—
241C2×D79—424no—
242C4×C79—22no—
243C4×C79—26no—
244C4×C79—22no—
245C4×C79—26no—
246C4×C79C22×#41112yes—
247C4×C79C2×H(5,𝔽2)14yes—
248Dic79—24no—
249Dic79—212no—
250Dic79—24no—
251C2×#3517—791nomaximal, Sylow
252C2×C79—46no—
253C2×C79#351714yescomplemented
254C2×C79C23×#4111yescharacteristic
255C2×C79C22×H(5,𝔽2)11yescharacteristic
256C2×C79C2×#58511yescharacteristic
257D79—412no—
258#3517—7932no—
259C2×#557—791no—
260C2×#553—796no—
261C2×#550—792no—
262C2×#550—791no—
263C2×#550—793no—
264C2×#547—793no—
265C2×#540—796no—
266C2×#532—793no—
267C2×#529—793no—
268C2×#520—792no—
269C2×C4×#41D7911yescharacteristic, complemented
270C79C2×#351711yescharacteristic, complemented, Sylow
271C22×#41—792no—
272C22×#41—796no—
273C22×#41C2×D7911yescharacteristic
274#557—7916no—
275#553—7996no—
276#550—7932no—
277#550—7916no—
278#550—7948no—
279#547—7948no—
280#540—7996no—
281#532—7948no—
282C2×C4⋊Q8—793no—
283C2×#127—796no—
284C2×#127—793no—
285#529—7948no—
286C2×#126—796no—
287C2×#125—796no—
288C2×#125—791no—
289C2×#124—793no—
290C2×#123—796no—
291C2×#123—7912no—
292C2×C22⋊Q8—793no—
293C2×#121—796no—
294C2×#121—793no—
295C2×#121—796no—
296C2×#121—796no—
297C2×#121—796no—
298C2×#121—7912no—
299#520—7932no—
300C4×#41C2×D79116yescomplemented
301C2×C4×Q8—792no—
302C2×C4×Q8—793no—
303C2×C4×Q8C2×D7911yescharacteristic
304C2×C4×Q8C2×D7911yescharacteristic
305C2×C4×D4—796no—
306C2×C4×D4—796no—
307C2×C4×D4—796no—
308C2×C4×D4—793no—
309C2×C4×D4—796no—
310C2×C4×D4C2×D7913yes—
311C2×C4×D4C2×D7913yes—
312C2×#117—796no—
313C2×#117C2×D7913yes—
314C22×C4⋊C4—796no—
315C22×C42C2×D7913yes—
316C2×#41—28no—
317C2×#41—31624no—
318C2×#41—7916no—
319C2×#41—7948no—
320C2×#41—15816no—
321C2×#41C22×D7918yes—
322C22×Q8—793no—
323C22×Q8C22×D7911yescharacteristic
324C22×D4—796no—
325C22×D4—7912no—
326C22×D4—796no—
327C22×D4—796no—
328C22×D4—793no—
329C22×D4C22×D7913yes—
330C23×C4—7912no—
331C23×C4—796no—
332C23×C4C22×D7913yes—
333C23×C4C22×D7913yes—
334C23×C4C22×D7913yes—
335C4⋊Q8—7924no—
336#127—7948no—
337#127—7924no—
338#126—7948no—
339#125—7948no—
340#125—798no—
341#124—7924no—
342#123—7948no—
343#123—7996no—
344C22⋊Q8—7924no—
345#121—7948no—
346#121—7924no—
347#121—7948no—
348#121—7948no—
349#121—7948no—
350#121—7996no—
351C4×Q8—7916no—
352C4×Q8—7924no—
353C4×Q8C22×D7918yes—
354C4×Q8C22×D7918yes—
355C4×D4—7948no—
356C4×D4—7948no—
357C4×D4—7948no—
358C4×D4—7924no—
359C4×D4—7948no—
360C4×D4C22×D79124yes—
361C4×D4C22×D79124yes—
362#117—7948no—
363#117C22×D79124yes—
364C2×C4⋊C4—796no—
365C2×C4⋊C4—796no—
366C2×C4⋊C4—792no—
367C2×C4⋊C4—791no—
368C2×C4⋊C4—792no—
369C2×C4⋊C4—796no—
370C2×C4⋊C4—791no—
371C2×C4⋊C4—796no—
372C2×C4⋊C4—7948no—
373C2×C4⋊C4C22×D7913yes—
374C2×C4⋊C4C22×D7913yes—
375C2×C22⋊C4—796no—
376C2×C22⋊C4—796no—
377C2×C22⋊C4—796no—
378C2×C22⋊C4—796no—
379C2×C22⋊C4—796no—
380C2×C22⋊C4C22×D7916yes—
381C2×C42—796no—
382C2×C42C22×D7913yes—
383C2×C42C22×D7913yes—
384C2×C42C22×D7913yes—
385C2×C42C22×D79124yes—
386C2×C42C22×D7911yescharacteristic
387C24—796no—
388C24C23×D7913yes—
389#41—232no—
390#41—31696no—
391#41—15864no—
392C2×Q8—7912no—
393C2×Q8—15812no—
394C2×Q8#1192962418yescomplemented
395C2×Q8C23×D7914yes—
396C2×D4—212no—
397C2×D4—31612no—
398C2×D4—31612no—
399C2×D4—31624no—
400C2×D4—7924no—
401C2×D4—7948no—
402C2×D4—7924no—
403C2×D4—7924no—
404C2×D4—7912no—
405C2×D4—15824no—
406C2×D4—15812no—
407C2×D4C23×D79112yes—
408C22×C4—212no—
409C22×C4—212no—
410C22×C4—46no—
411C22×C4—31624no—
412C22×C4—7948no—
413C22×C4—792no—
414C22×C4—796no—
415C22×C4—792no—
416C22×C4—7924no—
417C22×C4—15824no—
418C22×C4C23×D79112yes—
419C22×C4C23×D79112yes—
420C22×C4C23×D7911yescharacteristic
421C22×C4C23×D7911yescharacteristic
422C22×C4C23×D7913yes—
423C22×C4C23×D7913yes—
424C22×C4C23×D79112yes—
425C22×C4C23×D7913yes—
426C22×C4C23×D7911yescharacteristic
427C4⋊C4—7924no—
428C4⋊C4—798no—
429C4⋊C4—798no—
430C4⋊C4—7924no—
431C4⋊C4—7924no—
432C4⋊C4—794no—
433C4⋊C4—7924no—
434C4⋊C4—794no—
435C4⋊C4C23×D79112yes—
436C4⋊C4C23×D79112yes—
437C22⋊C4—7924no—
438C22⋊C4—7924no—
439C22⋊C4—7924no—
440C22⋊C4—7924no—
441C22⋊C4—7924no—
442C22⋊C4C23×D79124yes—
443C42—7924no—
444C42C23×D7914yes—
445C42C23×D79112yes—
446C42C23×D79112yes—
447C42C23×D79112yes—
448C23—26no—
449C23—26no—
450C23—26no—
451C23—7912no—
452C23—15812no—
453C23—15812no—
454C23—15812no—
455C23C24×D7916yes—
456C23C24×D7911yescharacteristic
457Q8—15824no—
458Q8C2×#11929624116yescomplemented
459D4—224no—
460D4—31624no—
461D4—31624no—
462D4—31648no—
463D4—15824no—
464D4—15848no—
465C2×C4—26no—
466C2×C4—22no—
467C2×C4—26no—
468C2×C4—22no—
469C2×C4—224no—
470C2×C4—224no—
471C2×C4—412no—
472C2×C4—31648no—
473C2×C4—794no—
474C2×C4—7912no—
475C2×C4—794no—
476C2×C4—15812no—
477C2×C4—1584no—
478C2×C4—1584no—
479C2×C4—15848no—
480C2×C4C2×#11929624112yes—
481C2×C4#1457920714yes—
482C2×C4C24×D7912yes—
483C2×C4C24×D7912yes—
484C2×C4C24×D7916yes—
485C2×C4C24×D7916yes—
486C2×C4C24×D7912yes—
487C2×C4C24×D7916yes—
488C22—26no—
489C22—26no—
490C22—26no—
491C22—412no—
492C22—31624no—
493C22—15812no—
494C22—15812no—
495C22—15812no—
496C22C22×#1192962412yes—
497C22C2×#1457920712yes—
498C22#1909939912yes—
499C22C25×D7911yescharacteristic
500C4—22no—
501C4—26no—
502C4—22no—
503C4—26no—
504C4—1584no—
505C4—15812no—
506C4—1584no—
507C4C22×#11929624112yes—
508C4C2×#1457920714yes—
509C2—46no—
510C2—31612no—
511C2#2377740614yesdirect product factor
512C2C23×#1192962411yescharacteristic
513C2C22×#1457920711yescharacteristic
514C2C2×#1909939911yescharacteristic
subgroupG/H
centerC23C24×D79
commutator, cosocleC22×C79C26
FrattiniC22C25×D79
FittingC2×C4×#41×C79C2
socleC23×C79C25
hypercenterC2×C4×#41D79
hypocenterC79C2×#3517
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C4×#41

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #23777406C2direct (N × H)
2. C2×#19099292C2semidirect
3. C2×#19099257C2semidirect
4. C2×#19099235C2semidirect
5. C2×#19099234C2semidirect
6. C2×#19099232C2semidirect
7. C2×#19099218C2semidirect
8. C2×#19099164C2semidirect
9. C2×#19099113C2semidirect
10. C2×#19099097C2semidirect
11. C2×#19099045C2semidirect
12. C2×C4×#41×C79C2semidirect
15. #19099292C22semidirect
16. #19099257C22semidirect
17. #19099235C22semidirect
18. #19099234C22semidirect
19. #19099232C22semidirect
20. #19099218C22semidirect
21. #19099164C22semidirect
22. #19099113C22semidirect
26. #19099097C22semidirect
40. #19099045C22semidirect
51. C4×#41×C79C22semidirect
167. C2×Q8×C79#41semidirect
199. Q8×C79C2×#41semidirect
253. C2×C79#3517semidirect
269. C2×C4×#41D79semidirect
270. C79C2×#3517semidirect
300. C4×#41C2×D79semidirect
394. C2×Q8#11929624semidirect
458. Q8C2×#11929624semidirect
511. C2#23777406direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#237774062
C2×#190992922
C2×#190992572
C2×#190992352
C2×#190992342
C2×#190992322
C2×#190992182
C2×#190991642
C2×#190991132
C2×#190990972
C2×#190990452
C2×C4×#41×C792
C2×#351779